AISM 53, 599-619

© 2001 ISM

## Distributions of numbers of success runs of fixed length in Markov
dependent trials

### Demetrios L. Antzoulakos and Stathis Chadjiconstantinidis

Department of Statistics and Insurance Science,
University of Piraeus, 80 Karaoli & Dimitriou Str., 18534 Piraeus, Greece, e-mail:dantz@unipi.gr

(Received January 11, 1999; revised August 9, 1999)

Abstract.
Let $\{ Z_n, n\geq 1\}$ be a time-homogeneous
$\{0,1\}$-valued Markov chain, and let $N_n$ be a random variable denoting the number of runs of "$1$" of length $k$ in the first $n$ trials. In this article we conduct a
systematic study of $N_n$ by establishing formulae for the evaluation of its probability generating function, probability mass function and moments. This is done in three different enumeration schemes for counting runs of length $k$, the "non-overlapping", the "overlapping" and the "at least" scheme. In the special case of i.i.d. trials several new results are established.

Key words and phrases:
Binomial/negative binomial distribution of order $k$, success runs, Markov chain,
probability generating function, probability mass function, moments.

**Source**
(TeX , DVI )